Thoughts on Olympiads

An olympiad in the context of this article is an academic competition in the form of a written exam paper. I will primarily be talking about the British Mathematics Olympiad.

My Own Experience with Maths Olympiads

My career in mathematical olympiads was short-lived and totally unremarkable.

I achieved gold awards on UKMT's maths challenges throughout secondary school, but only reached the olympiad level for the first time in year 12. I did try to get into the intermediate olympiads, but I always fell short, as I just wasn't very good at thinking during secondary school. Looking back, I'm sure that if I had done well enough to qualify, I would've struggled to answer more than a single question, but there is a question of "what if" I had tried harder. When I sat my first and penultimate maths olympiad paper, I was quite ill, and I did even worse than I had done in practice, where I usually couldn't even do the first and second questions. My embarrassing performance netted me a qualification certificate, and I could've swore it was pointing and laughing at me.

By this point, I had long since resigned that I'd never do "well" on an olympiad, but eleven months later, it was time for attempt 2 (of BMO1). I found just enough willpower to learn the basics of olympiad maths "properly", using the resources cited below, and I managed to answer some of the first three questions on some BMO1 papers while practising. I ended up with a score of 21/60, with ten points for each of the first two questions, zero for the next three, and just one for the froggy finale. Apparently the paper was significantly harder than those which preceded it, because the awards mark boundaries dropped to the point where I was able to redeem a certificate of distinction for 18 marks, down from the 26 needed in 2024.

I believe I've now earned just enough credit so that I can complain about maths olympiads are without completely looking like a sore loser.

The Hidden Specification of Maths Olympiads

In the UK, and seemingly across the whole world, maths olympiads exclude nearly all topics taught in the school curriculum, and basically all topics taught in A-Levels. At first, it might seem like this levels the playing field, and makes questions more about problem solving than learning meethods, but this isn't true. Olympiads do actually draw from a "specification", which isn't made public, so to do well, you have to spend time learning these new topics specifically, alongside your standard studies. This divides people who sit olympiad papers into two classes, those who don't learn from this "specification", and those who invest a large amount of time to learning the "specification".

A BMO1 paper will typically include at least one question on each of algebra, combinatorics, geometry, and number theory. The paper will typically start with relatively straightforward questions which you can immediately determine the topic of, and finish with problems whose required techniques are not at all obvious.

Evan Chen has great resources for mathematical olympiads, including this syllabus which covers the topics and techniques required by the American and British olympiads. Although I pesronally couldn't breach further than the starting chapters, his "textbook", the OTIS excerpts looks like a great place to start for those who want to take these competitions seriously for whatever reason.

Of the four topics, I actually quite like three of them. Algebra is an essential tool for most of mathematics, and I found the olympiad-specific content on the topic quite interesting. However, olympiads expect you to memorise theorems, which would be practically impossible to derive on the spot in the time-pressured exam. Going further, olympiads introduced me to functional equations, which are actually very fun, but frustratingly the British olympiads don't really use them.

For British olympiads, combinatorics is a relatively simple topic, but the questions were always too mystifying for me to even reach a stage where I could invoke combinations or permutations. I'll put the blame on myself for being stupid. Evan Chen's textbook stretches the topic to 90 densely packed pages, which horrifies me.

Number theory is another topic which is fun to learn about, but not fun in an olympiad paper. Geometry eludes me and literally everyone else I know, but from what I can tell, you have to memorise a lot of theorems and happen to "spot" their application on a question.

On any olympiad question, it's easy to go around in circles for a long time without spotting the "trick", and really, if the questions had just a tiny bit more structure, they'd be a lot more enjoyable.

The Physics Olympiad is just better

Olympiads also exist for other subjects. I can't speak much about olympiads for subjects like linguistics or chemistry, but I can for physics.

The British Physics Olympiad is similar to the maths olympiads, but just a lot better. I don't even like physics very much, but I found physics olympiad problems significantly more enjoyable and approachable than maths olympiad problems, even though I've spent more time with maths olympiads. I achieved a gold award on round 0 and round 1 of the physics olympiad in 2025, which wasn't quite enough to reach round 2, but was more than I had hoped for. "Round 0" is multiple choice, and serves as a filter to save examiners' time like the SMC does. The BPhO is quite brilliant, for the following reasons:

Content is based off school knowledge, without the expectation to learn an entirely new syllabus for the first rounds of the competition.
Content for round 0 and round 1 meets the standard of A-Level Physics, so students don't have to go back to the level they were studying two years ago.
Content meets the standard of A-Level Mathematics and A-Level Further Mathematics, so the physics olympiad actually involves a lot more familiar maths knowledge than the maths olympiads.
More advanced knowledge is required for round 2 and the international olympiad, following a natural progression from physics taught in school, rather than introducing completely new, random, and irrelevant topics like the maths olympiads do.
Round 1 features short-form questions and long-form questions each with variety in topics.
Test takers aren't required to answer all questions to obtain full marks, as there are more marks on the paper than the maximum number obtainable, meaning you don't have to put up with topics you hate or immediately unintuitive spotting the trick based questions.
Short-form questions and parts of long-form questions have different mark counts, so the harder and longer a question is, the more marks that are allocated to it. This allows for more variety in the paper and students' approaches to it.
The first short-form questions tend to be relatively easy and have appropriately low mark counts, making the paper more approachable to more candidates, while retaining the difficulty of scoring high marks overall.
Long-form questions are split into several structured parts, rather than giving the test taker a single question with nothing to work from.

If you rephrase each of these comments in a negative way, as if the competition took the exact opposite approach, you have my list of complaints for the maths olympiads. The most offensive thing about maths olympiads to me is that there's no calculus, just why!? The physics olympiad shows that it can be done. Why are maths olympiad organisers so against incorporating interesting, approachable, and useful topics? Combinatorics, algebra, geometry, and number theory are not that exciting to make them the focus of every olympiad paper for fifty years.

I'll never sit another olympiad in my life, as they are only for school students, so I'll likely have nothing more to write on the topic. That said, I might participate in "drunk BMO1" in the future.